Given a word rewriting system with variables $R$ and a word
with variables $w$ the question we are interested in is
whether all the instances of
$w$ obtained by substituting its variables by non-empty words
are reducible by $R$. This property is known as {\em ground
reducibility} and is the core of the {\em inductive
completion} methods that have been designed for proving theorems
in the initial model of equational specifications.
We prove the problem to be generally
undecidable
even for a very simple
word $w$, namely $axa$ where $a$ is a letter and $x$ a variable.
When $R$ is left-linear,
the question is decidable for arbitrary (linear or non-linear) $w$.